"It just shows the U.S. still lags behind other countries when it comes to an emphasis on educating American-born students in computer science, math and other areas," said Klein, who handles visa requests.

(Read more: Top court to hear Amazon workers on security checks)

Klein explained what's become a cycle for foreign nationals seeking jobs in the U.S.

"So many get advanced degrees at American universities that natural-born citizens don't receive, and (which) are needed for the hard-to-fill jobs," he said. "They go home, and yet so many of them make up the number of applications to work here."

It's not just the areas of technology and science that are seeing a need for foreign-born workers in the U.S., Klein added. He said he's processing applications for jobs in accounting, advertising and architecture.

Number of visa applications for companies to bring foreign born workers to the U.S. in 2013

Rank

H1B Visa Sponsor

Number of LCA

Average Salary

1

Infosys

15,810

$75,062

2

Wipro

7,178

$76,920

3

Tata Consultancy Services

6,732

$64,350

4

Ibm

6,190

$82,630

5

Deloitte Consulting

4,735

$98,305

6

Microsoft

4,067

$109,566

7

Larsen & Toubro Infotech

3,253

$59,241

8

Accenture

2,653

$72,704

9

Ernst & Young

2,316

$86,428

10

Satyam Computer Services

2,310

$70,495

Source: MyVisaJobs.com

Costly fees

To get an H-1B visa, the applications are placed into one of two computerized lotteries. Those with a masters degree are listed for the first 20,000 openings out of the total 85,000. If they are not selected, they fall into the main lottery for 65,000 spots—with a second chance for a job.

Application and lawyer fees, which must be paid by the company and not the applicant, can run from as low as $1.500 to more than a total of $5,000 each, according to analysts.

"Not every company plays by the rules, and that includes paying a certain wage," said Thompson & Knight's Klein. H-1B workers are supposed to receive fair compensation at the going rate for the job in the region of the work, but it doesn't always happen, explained Klein.